Lumber had been 3 less, they would have had a shilling apiece more to pay. What was their number, and what had each to pay? Ans. 24 persons; each paid 7 shillings. 14. A certain number consists of two places of figures, units and tens; the number is equal to 4 times the sum of its digits, and if 27 be added to the number, the order of the digits will be inverted. What is the number? NOTE 1.-Let x represent the digit in the place of tens, and y the digit in place of units; then 10x + y will express the number. Ans. 36. 15. A number is expressed by three figures whose sum is 11; the figure in the place of units is double that in the place of hundreds; and if 297 be added to the number, the result will be expressed by the same figures with their order reversed. What is the number? Ans. 326. 16. Divide the number 90 into three parts, such that twice the first part increased by 40, three times the second part increased by 20, and four times the third part increased by 10, may all be equal to one another. Ans. First part, 35; second, 30; third, 25. 17. A person placed $100000 out at interest, a part of it at 5 per cent., and the rest at 4 per cent.; the yearly interest received on the whole was $4640. Required the two parts of the principal. Ans. $64000 and $36000. 18. A person put out a certain sum of money at interest at a certain rate. Another person put out $10000 more than the first, at a rate per cent. greater by 1, and received an income greater by $800. A third person put out $15000 more than the first, at a rate per cent. greater by 2, and received an income greater by $1500. Required the three principals, and the respective rates of interest. NOTE 2.-To avoid the inconvenience of large numbers in the operation put a = 5000; then 2a = 10000, 3a 15000, θα 1500, and 16a = 800. I the final result, the value of a may be restored. Principals, $30000, $40000, $45000. Ans. Rates, 4, 5, 6, per cent. 19. If B's age be subtracted from A's, the difference will be C's age; if 5 times B's age and twice C's age be added together, and from their sum A's age be subtracted, the remainder will be 147; and the sum of the three ages is 96. Required the ages of A, B, and C, respectively. Ans. A's, 48; B's, 33; O's, 15. 20. Find what each of three persons, A, B, and C, is worth, knowing, 1st, that what A is worth added to 3 times what B and C are worth, is equal to 4700 dollars; 2d, that what B is worth added to 4 times what A and C are worth, is equal to 5800 dollars; 3d, that what C is worth added to 5 times what A and B are worth, is equal to 6300 dollars. Ans. A, $500; B, $600; C, $800. 21. A grocer sold 50 pounds of tea at an advance of 10 per cent. on the cost, and 30 pounds of coffee at an advance of 20 per cent. on the cost, and received for the whole $27.40, gaining $2.90. What was the cost per pound of the tea and coffee? Ans. Tea, $.40; coffee, $.15. 22. Five persons, A, B, C, D, E, play at cards; after A has won one half of B's money, B one-third of C's, C one-fourth of D's, D one-sixth of E's, they have each $30. How much had each to begin with? Ans. A, $11; B, 838; C, $33; D, $32; E, $36. 23. Three brothers desired to make a purchase, requiring $2000 of each. The first wanted, in addition to his own money, of the money of the second; the second wanted, in addition to his own, of the money of the third; and the third wanted, in addition to his own, of the money of the first. How much money d each? Ans. 1st, $1280; 2d, $1440; 3d, $1680. 24. A courier was sent from A to B, a distance of 147 miles; after 28 hours had elapsed, a second courier was sent from the Game place, who overtook the first just as he entered B. Now the time required by the first to travel 17 miles, added to the time required by the second to travel 56 miles, is 13 hours. How many miles did each travel per hour? Ans. 1st, 3 miles; 2d, 7 miles. 25. Find two numbers, such that if of the greater be added to of the less, the sum shall be 13; and if of the less be subtracted from of the greater, the remainder will be nothing. Ans. 18 and 12. 26. Find three numbers of such magnitudes, that the first added to of the sum of the other two, the second added to of the sum of the other two, and the third added to † of the sum of the other two, may each be equal to 51. Ans. 15, 33, and 39. 27. Said A to B and C, "If each of you will give me 4 sheep, I shall have 4 more than both of you will have left." Said B to A and C, "If each of you will give me 4 sheep, I shall have twice as many as both of you will have left." Said C to A and B, "If each of you will give me 4 sheep, I shall have three times as many as both of you will have left." How many sheep had each ? Ans. A, 6; B, 8; C, 10. 28. What fraction is that, to the numerator of which if 1 be added, the fraction will be ; but if to the denominator 1 be added, the fraction will be ? Ans. 29. What fraction is that, to the numerator of which if 2 be added, the fraction will be ; but if to the denominator 2 be added, the fraction will be ? Ans. . 30. Four persons, A, B, C, D, were engaged together in mowing for 4 successive days. The first day A worked 1 hour, B 3 hours, C 2 hours, and D 2 hours, and all together mowed 1 acre; the second day A worked 3 hours, B 2 hours, C 4 hours, and D 11 hours, and all together mowed 2 acres; the third day A worked 5 hours, B 4 hours, C 12 hours, and D 5 hours, and all together mowed 3 acres; the fourth day A worked 9 hours, B 7 hours, C 6 hours, and D 8 hours, and all together mowed 4 acres. How many hours would each alone require to mow 1 acre? Ans. A, 5 hours; B, 6 hours; C, 12 hours; D, 15 hours. 31. If A give B $5 of his money, B will have twice as much money as A has left; and if B give A $5, A will have thrice as much as B has left. How much has each ? Ans. A, $13; B, $11. 32. A corn factor mixes wheat flour, which cost him 10 shillings per bushel, with barley flour, which cost 4 shillings per bushel, in such a ratio as to gain 43 per cent. by selling the mix ture at 11 shillings per bushel. Required the ratio. Ans. The ratio is 14 bushels of wheat flour to 9 of barley. 33. There is a number consisting of two digits, which number divided by 5 gives a certain quotient and a remainder of 1, and the same number divided by 8 gives another quotient and a remainder of 1. Now the quotient obtained by dividing by 5 is twice the value of the digit in the tens' place, and the quotient obtained by dividing by 8 is equal to 5 times the digit in the units' place. What is the number? Ans. 41. 34. The four classes in a certain college are to compete for four prizes, amounting in the aggregate to $119, and the prize money is to be raised by contribution, on the following conditions, namely: that the members of the class whose candidate obtains the 1st prize shall each pay one dollar, and the class whose candidate obtains the 2d prize shall pay the remainder. Now it is found that if a senior gets the 1st prize and a junior the 2d, each junior will pay of a dollar; if a junior gets the 1st prize and a sophomore the 2d, each sophomore will pay of a dollar; if a sophomore gets the 1st prize and a freshman the 2d, each freshman will pay of a dollar; and if a freshman gets the 1st prize and a senior the 2d, each senior will pay of a dollar. Of how many members does each class consist? Ans. Freshman, 104; Sophomore, 93; 35. Find four numbers, such that if 3 times the first be added to the second, 4 times the second be added to the third, 5 times the third be added to the fourth, and 6 times the fourth be added to the first, each sum shall be 359. Ans. 95, 74, 63, 44. GENERAL SOLUTION OF PROBLEMS. 175. In the preceding problems, the given quantities have been expressed by numbers, and it has been required simply to determine the values of the unknown quantities from the numerical relations thus expressed. If, however, the given quantities in any problem be represented by letters, the solution will give rise to a formula, showing not only the value of the unknown quantity, but indicating the precise operations to be performed in order to obtain this valueThis is called a general solution of the problem. 176. When any particular problem has been proposed, we may, by simply varying the numbers, form other problems of the same kind or class; and the solutions of all the problems of the class will require exactly the same operations. Hence, 177. The General Solution of a problem is the process of obtaining a formula which shall express, in known terms, the values of the unknown quantities in the given problem, or in any problem of its class. 178. An Arbitrary Quantity is one to which any value may be assigned at pleasure, in a general formula or equation. 179. For illustration, let the following questions be proposed: 1. What number is that whose third part exceeds its fourth part by 6? Instead of confining our attention to the particular numbers here given, we may first investigate the problem under a general form, as follows: What number is that whose mth part exceeds its nth part by a? Let x represent the number; then by the conditions, Equation (3) is the formula which indicates the operations to be performed in solving all questions of this class. If in this formula we put m=3, n=4, and a=6, we shall have the number required by the particular question as at first proposed. 2. What number is that whose fifth part exceeds its seventh part by 12? To obtain the number by the formula, let m=5, n = 7, and a = 12; then x= 12 × 5 × 7 =210, Ans. |